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The Dodecahedron an Artistic Exploration Into Links Between Our 3D World and 2D Methods of Visualization

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The Dodecahedron an Artistic Exploration Into Links Between Our 3D World and 2D Methods of Visualization FORMLABS LESSON PLAN The Dodecahedron An artistic exploration into links between our 3D world and 2D methods of visualization. Adapted by Formlabs, original submission by Chris Sloan of Science Visualization. Formlabs’ Innovate & Educate Challenge invited educators across the country to develop and share lesson plans that encourage creative thinking and hands-on learning through 3D printing. Thanks to those who submitted, we’re able to share free resources with a wider learning community dedicated to inspiring students with rich, immersive experiences. Learn more about our growing library of lesson plans. Missed the challenge but have an innovative lesson plan you’d like to share? Find more information at formlabs.com/innovate-and-educate-challenge Lesson plan tested and submitted by: EDUCATOR ORGANIZATION Chris Sloan Science Visualization LOCATION CONTACT Cumberland, Maryland, USA [email protected] SUMMARY The Dodecahedron makes use of a 3D printed kit to introduce students to hands-on methods of translating three-dimensional objects into two-dimensional space. The lesson draws historical connections to notable mindsets and techniques around orthographic views, linear perspective drawings, dodecahedron geometry, the golden ratio, and the Fibonacci series. Students will journey from pre-Renaissance schools of thought, through how da Vinci’s contributions have shaped how we depict our world, to innovations enabled by 3D modeling advancements. OBJECTIVES • Learn about translating three-dimensional objects into two dimensions • Learn about Platonic solids • Learn about geometric ratios in nature and in design • Learn about projective geometry SUGGESTED AUDIENCE 6th – 10th grade students SUBJECT & INDUSTRY LINKS Applicable Subjects Applicable Industries mathematics engineering & product design art research & education engineering model making & entertainment formlabs.com FORMLABS LESSON PLAN The Dodecahedron 2 TOOLS & MATERIALS TO SHARE Form 2 & Resin Tape Computer PreForm Software .STL File * PER STUDENT / GROUP 3D Printed Paper & Acetate Pencil Marker Compass Ruler Dodecahedron Kit LESSON OVERVIEW 5 min Do Now Sketch Challenge Students will attempt to translate their view of a 3D printed dodecahedron into a paper sketch. 10 min Foundation Background Introduce students to the “rationalization of space” and its connection to many industries, including 3D modeling, animation, architecture, game design, fine arts, and engineering. 30 min Exploration Projections Using a 3D printed Dodecahedron Kit, guide students through drawing a more precise dodecahedron. Use the “projection box” to teach principles of orthographic views as a way of translating three-dimensional space. Golden Ratio Walk students through an exploration of the proportions (the Golden Ratio) inherent in a dodecahedron and their link to orthographic views. Lead students in the construction of the Golden Ratio and guide them to create one face of a dodecahedron (a pentagon). Perspective Drawing Explain the difference between orthographic projections and perspective drawings. Teach students how to use the Dodecahedron Kit to create a perspective drawing of a dodecahedron. 10 min Retrospective Connecting the Dots The class will share their creations and discuss other approaches for developing orthographic views of other objects. 5 min Closing Looking Forward Discuss the relevance of 3D concepts explored in this lesson, in understanding our world and in sparking innovations across many industries. * Download the .STL and .FORM files at:formlabs.com/lesson-plan-dodecahedron formlabs.com FORMLABS LESSON PLAN The Dodecahedron 3 ACTIVITY ONE 5 min Do Now 1. Distribute one 3D printed dodecahedron to each student (or student group). 2. Lead an interactive group discussion around the dodecahedron geometry. 3. Explain that the process of drawing objects is translating three-dimensional space into two-dimensional space. 4. Invite students to draw a dodecahedron, using the 3D printed model as a reference. Tips: Be encouraging, this is a challenging task. Considering hinting that you will be teaching them ‘tricks’ for sketching 3D shapes. 5. Give examples of why being able to translate three-dimensional geometries into two dimensions is an important skill. Dodecahedron Geometry and History Dodecahedrons are one of five Platonic solids identified 2,400 years ago by the Greek philosopher Plato. They are characterized by have same-length edges and same-shape faces, and are building blocks of our three-dimensional world. Dodecahedrons have 20 equal-length edges and 12 pentagonal faces. Other Platonic solids: cube, tetrahedron, octahedron, icosahedron. Other Paths to Explore: Plato, Platonic Solids, Archimedean Solids. formlabs.com FORMLABS LESSON PLAN The Dodecahedron 4 ACTIVITY TWO 10 min Foundation 1. Review students’ sketches and point out that the many edges and faces of a dodecahedron can make the shape confusing to draw. 2. Describe that Pre-Renaissance artists also had difficultly translating three-dimensions into two- dimensional space. Show and discuss Lorenzetti painting. 3. Show students two artists’ (da Vinci and Durer) drawings of dodecahedrons. 4. Discuss the inter-relatedness of geometry, natural science, and art using the collaboration between Pacioli and da Vinci as an example. 5. Introduce the concept of “rationalization of space” and “projective geometry” in the context of Durer’s “perspective machine.” Renaissance Impact The Renaissance (1300 - 1600) marked a transitional period for the understanding of how to represent 3D objects in 2D. Ambrogio Lorenzetti’s painting portrays an unrealistic perspective. Renaissance artists and mathematicians, however, worked together to understand and depict three-dimensional space. These artists included Leonardo da Vinci, who prepared drawings of Platonic solids for the mathematician Luca Pacioli. Similarly, Albrecht Durer was fascinated by the connection between geometry, natural science, and art. Artists like Durer discovered that 3D space can be “projected” onto 2D surfaces in a predictable way. His image of artists using a “perspective machine” illustrates one way to “rationalize space.” This was the origin of projective geometry, which is used in many fields of STEAM to this day. Other Paths to Explore: Lorenzetti, Pacioli, da Vinci, Durer. Geometry Natural Sciences Art Durer 1300 1400 1500 1600 Lorenzetti Pacioli da Vinci ACTIVITY THREE 30 min Exploration 1. Help students set up their Kits, with the dodecahedron in the middle and with an acetate sheet (“Picture Plane”) taped to a side of the cube (“Projection Box”) that is perpendicular to the table. Secure the Projection Box to the table with tape. 2. Instruct students to hold a permanent marker in their dominant hand, and the Eyepoint Stick in their other hand. 3. Have students look at the dodecahedron at eye level, while holding the Eyepoint Stick vertically and halfway between their eyes and the Picture Plane. 4. Instruct students to close one eye, align the top of the Eyepoint Stick with a vertex of the dodecahedron, and to carefully trace the dodecahedron on the Picture Plane. Projective Geometry is the process of translating a three-dimensional object onto a two-dimensional plane. Orthographic Views are one way that projective geometry is used. The view, or “projection” of the dodecahedron differs depending on the viewing angle. With the dodecahedron inside the Projection Box, there are three distinct parallel views - one per dimension. This explains why it takes three orthographic views to describe a three-dimensional object. Linear Perspective is another way that projective geometry is used. This method uses a conical projection, rather than a parallel projection for a 3D-to-2D translation. Linear perspective helps artists mimic what the human eye sees. Renaissance artists used both methods as tools for depicting three-dimensional space in ways that had never been possible before. formlabs.com FORMLABS LESSON PLAN The Dodecahedron 6 ACTIVITY THREE – CONTINUED 30 min Exploration 1. Review students’ conical projection drawings. 2. Explain to students that complete orthographic views should make it possible to create the object in physical form. Ask students how they would approach creating accurate orthographic views. 3. Explore the relationship between the Fibonacci Sequence and the Golden Ratio with students. 4. Ask students to group their Measuring Sticks by length. There are two small (s), three medium (m), and two large (l) sticks. 5. Use the Measuring Sticks to bring the Fibonacci Sequence and Golden Ratio to life. s + s = m is analogous to 1 + 1 = 2 6. Ask students what the next stick length in the sequence would be. s + m = l 7. Challenge students to find the Golden Ratio proportions in the dodecahedron. s = dodecahedron edge m = pentagonal face diagonal l = distance between pentagon centers 8. Point out some real-world examples and applications of the Golden Ratio. Golden Ratio The geometry of pentagons and dodecahedrons is linked to the Golden Ratio, a phenomena that Leonardo’s friend, Luca Pacioli, wrote a whole book about. For centuries, artists, mathematicians, and scientists have found this number very interesting, due to its abundant natural presence. Fibonacci Sequence The Golden Ratio is related to a special sequence of numbers, whereby you start with two numbers and each subsequent number equals the sum of the two previous numbers. e.g. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,… As the sequence progresses, the result of dividing
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